Question

An ideal gas is compressed isothermally until its pressure is doubled and then allowed to expand adiabatically to regain its original volume ($\gamma =1.4$ and $2^{-1.4}=0.38)$. The ratio of the final to initial pressure is

• A. $0.76:1$
• B. $1:1$
• C. $0.66:1$
• D. $0.86:1$

Answer 1

Answer: (A)

$0.76:1$

Let initial pressure, volume and temperature be $P$, $V$ and $T$ respectively.

$\therefore$ $P_2=2P$

For isothermal process : $P_2{V}_2=P_1{V}_1$

$\left(2P\right)V_2=PV$ $⟹V_2=\frac{V}{2}$

For adiabatic process : $P_2{V}_2^{\gamma }=P_3{V}_3^{\gamma }$ (Given $\gamma =1.4$)

$\Rightarrow 2P(\frac{V}{2}{)}^{1.4}=P_3{V}^{1.4}$

$\Rightarrow$ $2(\frac{1}{2}{)}^{1.4}=\frac{P_3}{P}$

$\Rightarrow 2\times 0.38=\frac{P_3}{P}$

$\Rightarrow \frac{P_3}{P}=0.76$

Thus, $P_3:P=0.76:1$